Surds and Rationalizing the Denominator

Surds appear everywhere in school maths, from Pythagoras' theorem to trigonometry and quadratic equations. Examiners expect answers in their simplest surd form with a rational denominator, so knowing these rules can save you marks on almost every paper. This guide explains surds from the ground up and shows you how to rationalize any denominator.

What Is a Surd?

A surd is an irrational root, a root that cannot be simplified to a whole number or fraction. For example, \(\sqrt{2}\), \(\sqrt{5}\) and \(\sqrt[3]{7}\) are surds, but \(\sqrt{9} = 3\) and \(\sqrt{\frac{4}{25}} = \frac{2}{5}\) are not.

Surds have non-terminating, non-repeating decimals (\(\sqrt{2} = 1.41421\ldots\)), so we keep them in surd form for exact answers.

The Key Rules

For positive numbers \(a\) and \(b\):

$$ \sqrt{ab} = \sqrt{a}\,\sqrt{b} \qquad \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \qquad \sqrt{a}\,\sqrt{a} = a $$

But be careful: \(\sqrt{a + b} \neq \sqrt{a} + \sqrt{b}\). For example, \(\sqrt{9 + 16} = 5\), while \(\sqrt{9} + \sqrt{16} = 7\).

Simplifying Surds

To simplify, find the largest perfect square factor and take its root outside.

$$ \sqrt{72} = \sqrt{36 \times 2} = 6\sqrt{2} $$

$$ \sqrt{75} = \sqrt{25 \times 3} = 5\sqrt{3} $$

Useful perfect squares to remember: 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144.

Adding, Subtracting and Multiplying Surds

Like surds have the same number under the root, and only like surds can be added or subtracted, just like collecting like terms in algebra.

$$ 3\sqrt{5} + 4\sqrt{5} = 7\sqrt{5} $$

To multiply, multiply the outside numbers together and the inside numbers together:

$$ 2\sqrt{3} \times 5\sqrt{6} = 10\sqrt{18} = 10 \times 3\sqrt{2} = 30\sqrt{2} $$

Worked Example 1: Simplify and Combine

Simplify \(\sqrt{50} + \sqrt{18} - \sqrt{8}\).

Step 1: Simplify each surd:

• \(\sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2}\)
• \(\sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2}\)
• \(\sqrt{8} = \sqrt{4 \times 2} = 2\sqrt{2}\)

Step 2: All are now like surds, so combine:

$$ 5\sqrt{2} + 3\sqrt{2} - 2\sqrt{2} = 6\sqrt{2} $$

Expanding Brackets With Surds

Expand just as you would with algebra. Two identities are especially useful:

$$ (a + \sqrt{b})^2 = a^2 + 2a\sqrt{b} + b $$

$$ (\sqrt{a} + \sqrt{b})(\sqrt{a} - \sqrt{b}) = a - b $$

The second one is the secret behind rationalizing. Multiplying a surd expression by its conjugate removes the roots completely.

Rationalizing the Denominator

Rationalizing means rewriting a fraction so the denominator has no surds.

Case 1: A Single Surd

Multiply the top and bottom by that surd:

$$ \frac{6}{\sqrt{3}} = \frac{6}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{6\sqrt{3}}{3} = 2\sqrt{3} $$

Case 2: A Binomial Denominator

If the denominator is \(a + \sqrt{b}\), multiply top and bottom by the conjugate \(a - \sqrt{b}\), and vice versa.

Worked Example 2: Rationalize a Binomial Denominator

Rationalize \(\dfrac{5}{3 - \sqrt{2}}\).

Step 1: Multiply top and bottom by the conjugate \(3 + \sqrt{2}\):

$$ \frac{5}{3 - \sqrt{2}} \times \frac{3 + \sqrt{2}}{3 + \sqrt{2}} $$

Step 2: The denominator becomes a difference of squares:

$$ (3)^2 - (\sqrt{2})^2 = 9 - 2 = 7 $$

Step 3: Expand the numerator: \(5(3 + \sqrt{2}) = 15 + 5\sqrt{2}\).

Answer:

$$ \frac{15 + 5\sqrt{2}}{7} $$

Worked Example 3: Surds on Top and Bottom

Simplify \(\dfrac{\sqrt{5} + \sqrt{3}}{\sqrt{5} - \sqrt{3}}\).

Step 1: Multiply by the conjugate \(\sqrt{5} + \sqrt{3}\):

$$ \frac{(\sqrt{5} + \sqrt{3})^2}{(\sqrt{5})^2 - (\sqrt{3})^2} $$

Step 2: Denominator: \(5 - 3 = 2\).

Step 3: Numerator:

$$ (\sqrt{5} + \sqrt{3})^2 = 5 + 2\sqrt{15} + 3 = 8 + 2\sqrt{15} $$

Step 4: Divide:

$$ \frac{8 + 2\sqrt{15}}{2} = 4 + \sqrt{15} $$

Worked Example 4: A Classic FSc/Matric Question

If \(x = 2 + \sqrt{3}\), find \(x + \frac{1}{x}\) and \(x^2 + \frac{1}{x^2}\).

Step 1: Find \(\frac{1}{x}\) by rationalizing:

$$ \frac{1}{2 + \sqrt{3}} \times \frac{2 - \sqrt{3}}{2 - \sqrt{3}} = \frac{2 - \sqrt{3}}{4 - 3} = 2 - \sqrt{3} $$

Step 2: Add:

$$ x + \frac{1}{x} = (2 + \sqrt{3}) + (2 - \sqrt{3}) = 4 $$

Step 3: Use the identity \(x^2 + \frac{1}{x^2} = \left(x + \frac{1}{x}\right)^2 - 2\):

$$ x^2 + \frac{1}{x^2} = 4^2 - 2 = 14 $$

Common Mistakes

• Writing \(\sqrt{a + b} = \sqrt{a} + \sqrt{b}\). This is false.
• Adding unlike surds, such as \(\sqrt{2} + \sqrt{3} = \sqrt{5}\). Also false.
• Multiplying only the denominator by the conjugate. You must multiply top and bottom.
• Not fully simplifying, e.g. leaving \(\sqrt{12}\) instead of \(2\sqrt{3}\).
• Getting the conjugate sign wrong. Change only the sign between the two terms.

Key Takeaways

• A surd is an irrational root; simplify it by taking out the largest perfect square factor.
• Only like surds can be added or subtracted.
• Rationalize a single surd by multiplying by that surd, and a binomial by its conjugate.
• \((a + \sqrt{b})(a - \sqrt{b}) = a^2 - b\) is the key to removing surds from denominators.

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